All Questions
5,576 questions
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Specific regularity in bipartite graphs
Let $G(A,B)$ be a bipartite graph with $|A| = |B| = n$, where $n$ is sufficiently large. The average degree of $G$ is $d = \frac{e(A,B)}{n}$, where $e(A,B)$ denotes the number of edges between sets $A$...
- 359
-4
votes
0
answers
24
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Instrumental Variable model prove inequality holds [closed]
very stuck on this proof for my homework, professor didn't really teach this concept as it was supposed to be "self-learning", so I'm not really sure where to start.
homework problem
13
votes
4
answers
2k
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The ten most fundamental topics in geometric group theory
What are the ten most fundamental topics in geometric group theory?
This is a pedagogical question prompted by the fact that I am teaching geometric group theory to undergraduates. They are expected ...
1
vote
1
answer
49
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Graph classes which have small edge k-cuts
I am interested in graph classes that have the following property: There exists a function $f(k)$ such that for every graph $G$ in the class, for every choice of $k$ vertices $v_1, \ldots, v_k$ in the ...
- 526
-3
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0
answers
64
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Can both conditions about vertex degrees hold true in a planar graph? [closed]
I am working on a problem about planar graphs and trying to understand if two statements can both be true at the same time.
The problem states that for any planar graph with at least 3 or more ...
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1
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0
answers
62
views
Bipartite Representation of a Directed Graph
I am working on a combinatorial optimization problem and have constructed a bipartite graph as a representation of a directed graph.
The construction is as follows:
Given an initial directed graph $G$ ...
7
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0
answers
220
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Is there a Cayley graph with end space infinite and discrete?
A Cayley graph of a finitely generated group must be locally finite, and we know end spaces of locally finite graphs must be compact - so we can't have an infinite and discrete end space in this ...
- 407
5
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3
answers
292
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The max-clique chromatic number of a graph
Let $G = (V,E)$ be a graph. Every clique, that is, complete subgraph, is
contained in a maximal clique with respect to $\subseteq$ (this is
an easy consequence of Zorn's Lemma). Let $\newcommand{\MC}{\...
- 48.9k
1
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0
answers
126
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Growth polynomial of the Associahedron graph ? (Is it approximately Gaussian ?)
Consider Associahedron, consider graph build from its vertices and edges. Choose some vertex. Let us count the number of vertices on distances $k$ from the selected vertex. Write a generating ...
- 24.9k
0
votes
0
answers
37
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separator and vertex-connectivity
A definition of "separator" is the following: Let $G$ is an $n$-vertex graph, then $S\subseteq V(G)$ is a separator if there is a partition $V=A\cup B\cup S$ such that $|A|,|B|\le 2n/3$ and ...
- 281
3
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1
answer
133
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Is a simply connected locally 2-connected complex a union of spheres and planes?
Let $X$ be a (potentially infinite) 2-dimensional simplicial complex. Then each link at a vertex $x\in X$ is a graph.
Question. If $X$ is simply connected and each link is 2-connected (in the sense ...
- 13.6k
1
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1
answer
80
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What are the efficient algorithms to compute Hamiltonian paths on Cayley graphs of finite groups ? Can GAP do it?
The famous Lovasz conjecture predicts existence of the Hamiltonian path on Cayley graphs. In general finding such a path is NP-complete problem, but there are many heuristic algorithms.
Question 1: ...
- 24.9k
3
votes
1
answer
202
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Matrix-tree theorem for inverse matrices
Let $L$ be the Laplacian of a directed weighted graph on $n$ nodes, e.g., for $n=4$:
$$
L = \left(\begin{array}{cccc} w_{1,1}+w_{1,2}+w_{1,3}+w_{1,4} & -w_{1,2} & -w_{1,3} & -w_{1,4}\\ ...
- 20.2k
16
votes
1
answer
977
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Pedagogically intuitive reformulation of Zorn's Lemma for functional analysis
While teaching an applied functional analysis class, I’ve noticed that students often struggle to develop an intuitive understanding of Zorn’s lemma. It’s relatively straightforward to explain why ...
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2
votes
1
answer
111
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Is there a ternary Cayley graph on 27 vertices that is a non-complete core?
Is there a non-complete ternary Cayley graph that is a core with $3^3 = 27$ vertices?
By a ternary Cayley graph, I mean a (simple, undirected) graph whose vertex set is $\mathbb{Z}_3^n := \bigoplus_{i ...
- 331
1
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0
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237
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Claimed proofs of graph labelling conjectures [closed]
The following recent series of arXiv papers claims to prove several of the most famous graph labelling conjectures. Edinah Gnang is the common author, none of the papers seem to be published further, ...
- 1,319
0
votes
0
answers
24
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Minimizing intersections between spanning trees of graph embeddings in polynomial time
Assume I have $N$ complete graphs $G_1, G_2,...,G_N$, and consider their embeddings $E_1, E_2,...,E_N$ in $\mathbb{R}^2$. Is there a (potentially stochastic) polynomial time algorithm to construct ...
- 1
1
vote
1
answer
73
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"Gray code" for $[\omega]^{<\omega}$
Let $\newcommand{\oo}{[\omega]^{<\omega}}\oo$ denote the collection of finite subsets of the set of non-negative integers $\newcommand{\o}{\omega}\o$.
If $A,B$ are any sets, let $A \,\triangle \, B ...
- 48.9k
1
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1
answer
100
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Is there any known upper bound for the local crossing number of a graph drawing in the plane?
The local crossing number ${\rm LCR(G)}$ of a graph $G$ is defined as the least nonnegative integer $k$ such that the graph has a $k$-planar drawing. In other words, it is the smallest possible number ...
- 1,190
4
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0
answers
69
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is a 4-connected planar graph still Hamiltonian after removing an edge?
We know that 4-connected planar graphs are Hamiltonian(by the known Tutte Theorem). Additionally, Thomas and Yu [1] proved that removing two vertices from a 4-connected planar graph still preserves ...
- 1,851
0
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1
answer
51
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Cycle-Sculpturing with Minimal Vertex-Deletion
given a simple, finite and symmetric graph $G(V,E)$ with $n$ vertices and at least $n$ edges
Question:
how can the smallest set of vertices $W\subset V$ be calculated for which the graph induced by $...
- 13.2k
1
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0
answers
51
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Coarse-graining a hypergraph
$\DeclareMathOperator{\poly}{\mathrm{poly}}$I have asked this question on math.SE here, but couldn't get a satisfactory answer. I have also asked a related question on math overflow here, but haven't ...
- 277
1
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1
answer
87
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Bounds on the number of proper 3-colorings of cubic graphs
Are there known bounds on the number of proper 3-colorings of a 3-regular in terms of vertex count?
- 83
1
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0
answers
41
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Asymptotic mixing time and Euclidean probability distance for path graphs
We are given a simple path graph $P(V,E)$ with vertex set $V$ and edge set $E$, having $n=|V|$ nodes. Given an initial distribution $\mathbf{\mu}$ over $V$, let $d_t(\mathbf{\mu},\pi)$ be defined as $\...
- 1,677
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0
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42
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How to determine if two matchings are related by a permutation?
Let $n \geq 2$ be an integer. Let
\begin{align*}
V &= \{(i, j); 1 \leq i, j \leq n \text{ and } i \neq j \} \\
E &= \{ \{v_1, v_2\}; v_1, v_2 \in V \text{ and } v_1 \neq v_2 \}.
\end{align*}
...
- 5,215
0
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0
answers
57
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Does Forcing conjecture equals to assume the host graph is regular?
Given two graphs $H$ and $G$, the homomorphism density $t(H, G)$ is defined as the proportion of mappings from the vertices of $H$ to the vertices of $G$ that preserve adjacency. Formally,
$$
t(H, ...
- 359
3
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0
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93
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Realized graph of majority of permutations
This question was asked several months ago on Math.SE, but remains unsolved.
For any collection of permutations of $\{1,2,\dots,n\}$, we say that it realizes a directed multigraph with $1,2,\dots,n$ ...
- 277
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0
answers
51
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Subgraphs of random graphs with a given degree sequence
Let $\mathbf{d}=(d_1,\dots, d_n)$ be a given degree sequence with $3\leq d_i\leq \Delta$ for every $i$, where $\Delta$ is constant. Let $G(n,\mathbf{d})$ denote the random graph uniformly distributed ...
- 143
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1
answer
78
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Bipartite matching where every adjacent pair of vertex on the left side of the graph has at least 1 vertex matched
The question is as stated above. I want to devise bipartite matching algorithm where it determies whether every adjacent pair of vertex on the left side of the bipartite graph has at least 1 vertex ...
0
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0
answers
45
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Another version of Sidorenko's conjecture(?)
I would like to ask a question about Sidorenko's conjecture. Here is the background of my question:
Quasi-random graphs
A sequence of graphs $(G_n)$ is called quasi-random if it satisfies certain ...
- 359
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0
answers
36
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Construct a maximum matching from a minimum vertex cover in bipartite graph?
Konig's theorem in graph theory says that for a bipartite graph $G$, the size of maximum matching in $G$ is equal to the size of minimum vertex cover of $G$.
Typically, one of the proofs is to ...
- 281
2
votes
0
answers
130
views
Does Ising partition function determine the number of $k$-matchings mod $4$ for cubic graphs?
Let $G$ be a cubic graph. It's known that the Tutte polynomial $T_G$ of $G$ on the hyperbola $(x-1)(y-1)=2$ determines the Ising partition function of $G$ and vice versa.
A $k$-matching in a graph $G$ ...
- 9,501
1
vote
1
answer
90
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Characterizing the family of maximal cliques of a cograph
Preamble #1
There are two common equivalent definitions of cographs:
the smallest class that includes $K_1$ and is closed under disjoint union and complementation (or join);
the finite $P_4$-free ...
- 113
2
votes
0
answers
173
views
How many maximal length snakes are there?
This problem was motivated by the classic phone game Snake.
Consider the square grid graph with vertex set $V := \{1, \dots, N\}^2$, for fixed odd positive integer $N$, and an edge between $(x, y)$ ...
- 6,285
5
votes
1
answer
270
views
Approximation of Hamiltonian cycles
Let's define the $\texttt{MinHalfSimpCycle}$ search problem: Given $G=(V, E)$ a complete, undirected graph with weights on the edges. We want a simple cycle in $G$ (each vertex appears in it at most ...
- 53
2
votes
4
answers
212
views
Efficient algorithm for graph problem
Let $D=(V,E)$ be a directed graph, $S,T\subset V$ and $f:V\rightarrow \{1,\ldots, k\}$ a positive, bounded weight-function and $l\in \mathbb{N}$, find a path $v_1,\ldots, v_l\in V$ with $v_1\in S$ and ...
- 157
4
votes
1
answer
141
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Group generated by "adjacent" permutations of graph
Let $G=(V,E)$ be any graph, i.e. $E$ is simply a binary relation over $V$. We say a permutation $\sigma\in \text{Bij}(V)$ of the vertices of $G$ is adjacent if, for all $v\in V$, $(v,\sigma(v))\in E$. ...
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92
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Definition of Loop in an Oriented Matroid
I had posted this on Stackexchange because I don't believe this is a particlarly difficult question, but there were no answers, so I'm posting it on here now.
I just had a quick question about the ...
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29
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Explicit Bound in Draganić's Hamiltonicity Result?
Earlier this year, Draganić et al published a remarkable piece of work that resolved Krivelevich and Sudakov's conjecture on the Hamiltonicity of expanders. Here's the abstract:
An n-vertex graph G ...
- 3,979
4
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1
answer
252
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What is the resistance between two vertices on the Hanoi-towers graph?
The Hanoi graph $H^n_k$ is the graph with nodes representing states of a Hanoi puzzle with $n$ discs and $k$ pegs, and edges representing the various moves of the discs from peg to peg.
The Hanoi ...
- 2,185
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0
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50
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The uniqueness of a solution Impossible chessboard puzzle
This is a double post.
https://math.stackexchange.com/questions/4981887/the-uniqueness-of-a-solution-impossible-chessboard-puzzle/4981946#4981946
I found a video about a famous puzzle called "...
- 328
2
votes
2
answers
210
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Rank of adjacency matrix of a graph on a sphere all of whose faces have four vertices
Let $G$ be a graph drawn on the sphere such that every face of $G$ has exactly four vertices. Question: can anything be said about the rank of the adjacency matrix of $G$ in terms of other (preferably ...
- 2,964
8
votes
4
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1k
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Counting with trees
Let $\mathcal{U}_n$ denote the set of unrooted unlabelled trees with $n$ edges. For $T\in\mathcal{U}_n$, let $1^{u_1}2^{u_2}\cdots n^{u_n}$ be its degree distribution, that is, $u_i=\#$ of vertices ...
- 43.2k
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1
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127
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Petersen graph does not have a nowhere-zero 4-flow
I try to prove that the Petersen graph does not have a nowhere-zero 4-flow (i.e., over $\mathbb{Z}_4$), but I don't know how a proof could work...
I'm happy about every hint, thank you in advance!
3
votes
0
answers
51
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Asymptotic dimension of graph families representing each finite group
Frucht's theorem says every finite group is isomorphic to the automorphism group of a simple graph $G$ (with no loops, multiple edges or directed edges).
There has been interest in finding classes of ...
- 1,936
2
votes
1
answer
100
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Clique number and a special partition
Let $G=(V,E)$ be a finite, simple, undirected, connected graph, and let $\omega(G)$ denote its clique number. Assume that $G$ has a partition into $m$ independent subsets $U_1,\dots, U_m$ such that ...
- 21
2
votes
1
answer
226
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Expanders except for commutativity?
What would you call a graph that is an expander except for commutativity, in the following sense?
Say that, from every vertex, you have $d$ edges ($d$ large) labelled $x_1,\dotsc, x_d$. Say that your ...
- 20.2k
2
votes
0
answers
35
views
Limiting spectral distribution of a random matrix with specific structure
First, consider an $N \times N$ Hermitian random matrix $V$ from the Gaussian Unitary Ensemble (GUE). It is well known that the empirical spectral distribution of the GUE satisfies the semicircle law ...
- 21
0
votes
0
answers
25
views
Is there a name for a spanner graph that only considers distance to a root node?
A $t$-spanner graph of a set of points $\{p_i\}$ in the plane is a graph $G = (V, E)$ such that for any pair of vertices $p_i, p_j \in V$, the shortest path distance $d_G(p_i, p_j)$ in $G$ is at most $...
- 4,049
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0
answers
45
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Functional inequalities on neighbourhood graphs
Consider an open domain $\Omega \in \mathbb{R}^d$, say the unit disk in $\mathbb{R}^2$ with $N$ points sampled i.i.d. on it. One of the simplest possible (unnormalised) discrete Laplacian of a ...
- 111